On one-dimensional Riccati diffusions

Bishop, Adrian N.; Moral, Pierre Del; Kamatani, Kengo; Rémillard, Bruno

doi:10.1214/18-aap1431

Cited by 17 publications

(69 citation statements)

References 76 publications

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“…certain nonlinear diffusion models) and the convergence (with sample size) of particular EnKF methods. Time-uniform fluctuation, stability and contraction estimates were given in the one-dimensional linear-Gaussian case in [11]; with essentially no further assumptions on the underlying state-space model. In the multi-dimensional linear-Gaussian case, time-uniform stability estimates were developed under strong signal stability assumptions in [22].…”

Section: Ensemble Kalman-bucy Filtersmentioning

confidence: 99%

A perturbation analysis of stochastic matrix Riccati diffusions

Bishop¹,

Moral²,

Niclas³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman-Bucy filters since they describe the flow of certain sample covariance estimates. In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional. The main purpose of this article is to derive non-asymptotic Taylor-type expansions of stochastic matrix Riccati flows with respect to some perturbation parameter. These expansions rely on an original combination of stochastic differential analysis and nonlinear semigroup techniques on matrix spaces. The results here quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. The convergence of the interacting sample covariance matrices to the deterministic Riccati flow is proven as the number of particles tends to infinity. Also presented are refined moment estimates and sharp bias and variance estimates. These expansions are also used to deduce a functional central limit theorem at the level of the diffusion process in matrix spaces.Résumé: Les équations de Riccati matricielles jouent un rôle important dans la théorie du filtrage et du contrôle optimal. Cet article présente une théorie des perturbations d'une classe d'équations de Riccati matricielles stochastiques. Ces modèles probabilistes sont d'un usage courant dans la théorie des filtres de Kalman d'Ensemble. Ils représentent dans ce contexte l'évolution des matrices de covariance empiriques associées à un ensemble de diffusions en interaction. Les perturbations aléatoires résultent des fluctuations stochastiques d'un système de particules de type champ moyen interagissant avec la mesure empirique du système. Nous présentons dans cet article une formule de Taylor non asymptotique pour des flots stochastiques de diffusion de Riccati matricelles par rapport à un paramètre de fluctuation. Ces développements sont fondés sur un nouveau calcul différentiel stochastique et une analyse fine de semigroupes non linéaires dans des espaces de matrices. Ces résultats permettent de quantifier avec précision les fluctuations des flots de matrices stochastiques autour des systèmes limites à tout ordre. Nous illustrons ces résultats avec une preuve de la convergence des matrices empiriques de filtres de Kalman d'Ensemble vers la solution d'équations de Riccati déterministes lorsque le nombre de particules tends vers l'infini. Nous présentons dans ce cadre des estimations fines des biais et des variances, ainsi qu'un theorème de la limite centrale fonctionnel au niveau du processus matriciel.

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Section: Ensemble Kalman-bucy Filtersmentioning

confidence: 99%

A perturbation analysis of stochastic matrix Riccati diffusions

Bishop¹,

Moral²,

Niclas³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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“…by studying the stability properties of ensemble Kalman-Bucy-type filters [21]. See [9,8,7] for dedicated technical discussions on related ensemble filters, and associated literature review (which is beyond the focus of this article). Related applications of our type of stability result are in [25].…”

Section: An Illustrative Example: Ensemble Kalman-bucy Filtersmentioning

confidence: 99%

“…Moreover, there exists a unique matrix P ∞ ≥ 0 such that Ricc(P ∞ ) = 0 and the spectral abscissa of the matrix A ∞ := A − P ∞ S is negative (7) If (A, R 1/2 ) is controllable, then P ∞ > 0. When the pair (A, R 1/2 ) is stabilisable, and (A, S 1/2 ) is detectable, we have P t −→ t→∞ P ∞ and thus A t −→ t→∞ A ∞ exponentially fast for any initial matrices (8) For proof of these classical results on the true Kalman-Bucy filter see, e.g., [33,36], and the convergence results in [35,15,56,5,6]. Going forward we may assume an even stronger notion of controllability and observability; and suppose that the logarithmic norm (defined later) of A ∞ is negative.…”

Section: An Illustrative Example: Ensemble Kalman-bucy Filtersmentioning

confidence: 99%

“…One advantage of the deterministic-type ensemble Kalman filters [48], is that V = 0; so the sample covariance P ǫ t exhibits less fluctuation around its limiting value P t than in the vanilla case [21]. For details on these models and their fluctuation properties see [9,8,7] and the references therein.…”

Section: An Illustrative Example: Ensemble Kalman-bucy Filtersmentioning

confidence: 99%

“…As an illustrative example of our general stability results, we will relate our analysis of random linear diffusions to those associated with the ensemble Kalman-Bucy filter, and the drift and diffusion matrices (4). Related analysis in one-dimension is given in [8] where stronger results are available.…”

Section: An Illustrative Example: Ensemble Kalman-bucy Filtersmentioning

confidence: 99%

See 2 more Smart Citations

Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients

Bishop¹,

Moral²

2019

SIAM J. Control Optim.

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This work is concerned with the stability properties of linear stochastic differential equations with random (drift and diffusion) coefficient matrices, and the stability of a corresponding random transition matrix (or exponential semigroup). We consider a class of random matrix drift coefficients that involves random perturbations of an exponentially stable flow of deterministic (time-varying) drift matrices. In contrast with more conventional studies, our analysis is not based on the existence of Lyapunov functions, and it does not rely on any ergodic properties. These approaches are often difficult to apply in practice when the drift/diffusion coefficients are random. We present rather weak and easily checked perturbation-type conditions for the asymptotic stability of time-varying and random linear stochastic differential equations. We provide new log-Lyapunov estimates and exponential contraction inequalities on any time horizon as soon as the fluctuation parameter is sufficiently small. These seem to be the first results of this type for this class of linear stochastic differential equations with random coefficient matrices.1 The "random" predicate is used to denote the randomness of the coefficients. In the case of deterministic coefficients, we term this equation: a stochastic differential equation, or an Ito-type (linear) diffusion process (driven by a Wiener process). Note that Ito-type integrals do not preclude (suitably adapted) random integrands; however, our terminology is used to distinguish those equations with random coefficients and those without.2 "Random" is again used to describe the randomness of the coefficients and we refer to this process as an Ornstein-Uhlenbeck process since we are concerned ultimately with stable processes in an Ornstein-Uhlenbeck-type form.

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On the mathematical theory of ensemble (linear-Gaussian) Kalman–Bucy filtering

Bishop

Moral

2023

Math. Control Signals Syst.

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The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman–Bucy filtering for continuous-time, linear-Gaussian signal and observation models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble filtering. We consider these equations and their characteristics in a number of popular ensemble Kalman filtering variants. Given these equations, we study their asymptotic convergence to the optimal Bayesian filter. We also study in detail some non-asymptotic time-uniform fluctuation, stability, and contraction results on the sample covariance and sample mean (or sample error track). We focus on testable signal/observation model conditions, and we accommodate fully unstable (latent) signal models. We discuss the relevance and importance of these results in characterising the filter’s behaviour, e.g. it is signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman–Bucy filtering. We also provide a novel (and negative) result proving that the bootstrap particle filter cannot track even the most basic unstable latent signal, in contrast with the ensemble Kalman filter (and the optimal filter). We provide intuition for how the main results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence.

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On one-dimensional Riccati diffusions

Cited by 17 publications

References 76 publications

A perturbation analysis of stochastic matrix Riccati diffusions

A perturbation analysis of stochastic matrix Riccati diffusions

Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients

On the mathematical theory of ensemble (linear-Gaussian) Kalman–Bucy filtering

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